The Department of Mathematics Colloquia are normally held in 4010 University Hall on Fridays at 3:00pm. Any departures from this are indicated below.
Light refreshments are served after the colloquia in 2040 University Hall.
Driving directions, parking information, and maps are available on the university website.
What follows is a list of speakers, talk titles and abstracts for the current academic year. Abstracts for the talks are also posted in the hallways around the departmental offices.
Spiro Karigiannis (University of Waterloo, Canada)
An exceptional structure in 7-dimensional geometry
Abstract: For any fixed algebraic structure on a real vector space, one can consider a smoothly varying family of such structures on the tangent spaces of a manifold. In particular the exceptional algebraic structure of the octonions naturally leads one to define $G_2$ manifolds and their distinguished minimal submanifolds and Yang-Mills connections. The subject of $G_2$ geometry involves an interplay of non-associative algebra, differential geometry, and non-linear global analysis. I will present an introduction to $G_2$ manifolds for a general audience, paying particular attention to the similarities and differences of $G_2$ geometry with respect to the geometries of Kähler manifolds and of oriented Riemannian 3-manifolds. If time permits I will briefly survey some of the recent developments in the field, and discuss some of the important open problems for the future.
Vladimir Peller (Michigan State University)
New Development of Perturbation Theory
Abstract: I am going to speak about new results obtained jointly with A. B. Aleksandrov. It is known that a Lipschitz function $f$ (i.e., a function satisfying $\vert f(x)-f(y) \vert \le$ const $\vert x-y \vert$) on the real line does not have to be operator Lipschitz (i.e., $\Vert f(A)-f(A) \Vert \le$ const $\Vert A-B \Vert$ for all self-adjoint operators $A$ and $B$). We have proved that the situation changes dramatically if we consider the class of Hölder functions of order $\alpha$, $0<\alpha<1$. Such functions must be operator Hölder of order $\alpha$. We also prove similar results for other classes of functions. Then I am going to speak about the problem of what can be said about operators of the form $f(A)-f(B)$ if we know that $A-B$ belongs to a Schatten-von Neumann class $S_p$ for various classes of functions $f$.
Rafael López (Universidad de Granada, Spain)
Mean curvature,Physics, PDEs
Abstract: In physical phenomenon related with capillarity, interfaces are modeled by surfaces whose mean curvature satisfies a certain equation. The simplest case is that the mean curvature is a constant function (for example, in microgravity environment). The purpose of the seminar is to show different scenes and the possible geometric configurations that the surfaces can adopt. We discuss results and we introduce the techniques, specially those that come from the PDEs theory.
Gabriel Prajitura (SUNY at Brockport)
Linear dynamics
Abstract: We will discuss the geometric properties of orbits of linear operators in Hilbert spaces, and will analyze an attempt of extending the notion of density of such an orbit to the case of non-separable spaces.
Arshak Petrosyan (Purdue University)
Monotonicity Formulas and the Thin Obstacle Problem
Abstract: We will talk about some recent advancements in the thin obstacle (or Signorini) problem that were obtained by using several types of monotonicity formulas, including Almgren's frequency formula.
We will give a brief overview of the known results and then present some new results about the structure of the singular set.
This is a joint work with Nicola Garofalo.
Biao Wang (University of Toledo)
Foliations for Quasi-Fuchsian 3-Manifolds
Abstract: In this talk, we will show that if a quasi-Fuchsian 3-manifold contains a minimal surface whose principle curvatures are in $(-1,1)$, then it admits a foliation such that each leaf is a surface of constant mean curvature. The key method that we use here is volume preserving mean curvature flow.
Elizabeth Wulcan (University of Michigan)
Residues and ideals of holomorphic functions
Abstract: Multidimensional residue theory is a useful tool in the study of many questions concerning ideals of polynomials or holomorphic functions.
I will discuss an approach of associating a residue with an ideal of holomorphic functions, which extends classical constructions for complete intersection ideals. In particular, this residue generalizes the usual one variable residue of a meromorphic function. This talk will be based on joint work with Mats Andersson.
Ivan Sterling (St Mary's College of Maryland)
Constant Mean Curvature Surfaces after Wente
Abstract: In 1984 Wente solved a long standing problem in differential geometry: Is the round sphere the only compact constant mean curvature surface (without boundary)? Wente proved the answer was no, by proving the existence of cmc tori. Wente's work was the starting point for research, by many authors, in many different directions. It wouldn't be an exaggeration to say that a new branch of differential geometry has developed. Besides being of interest in its own right, this new branch intertwines with many different areas of mathematics and physics. We will focus on only a small part of this branch, giving a partial survey of results in the area of constant mean curvature surfaces.
Akaki Tikaradze (University of Toledo)
Representations of certain deformations of the enveloping algebras of Lie algebras
Abstract: I will discuss representation theory of certain associative algebras which are similar in spirit to the enveloping algebras of semi-simple Lie algebras. Particular attention will be paid to the case of positive characteristic of the ground field.